The challenge though is to get the two quantities εm and εH . We now have three equations and five unknowns, namely, u, v, T , εm , and εH . This is the fundamental problem of closure in turbulence modeling. We need to relate εm and εH to the average quantities u, v , or T so that we can effectively handle the problem of closure. This activity is actually a separate field by itself and is known as turbulence modeling.

A quick look at the qualitative nature of the velocity profiles in laminar and turbulent boundary layers is in order. The turbulent velocity profile is sharp, with a steep slope at y = 0, denoting higher shear stress at the wall (see Fig. 5.7). Discerning readers will quickly realise that the higher shear stress at the wall will lead to more heat transfer. While the former is not desirable, the latter is.

The ratio of εm to εH is known as turbulent Prandtl number denoted by PrT . Similar to being nearly equal to in gases, εm /εH is nearly equal to 1.

For Pr >>1 fluids, based on boundary layer measurements, the recommended value of PrT is around 0.9. The idea of a turbulent Prandtl number, which is strictly not a physical property and is apparently intangible, is a useful construct in heat transfer engineering, as from five unknowns, we have eliminated one.

Reynolds analogy

Let us now see if we establish a relationship between momentum transfer and heat transfer. If we are successful in doing this, then by performing the easier-to-do velocity measurements together with the use of Newton’s law of viscosity, we can

FIGURE 5.7

Figure showing a qualitative variation of horizontal velocity in (A) laminar and (B) turbulent boundary layers over a flat plate.

150 CHAPTER 5  Forced convection

From boundary layer measurements, for turbulent flow over a flat plate, the following critical information is available.

δ0.37

x= Rex0.2

With the above expression, using the modified Reynolds analogy, the local Nusselt number can be determined to be

Again, this is valid for fluids that satisfy 0.6 ≤ Pr ≤ 60

If the flow is mixed, that is, laminar up to x = xcrit and turbulent beyond, the average heat transfer coefficient hL is given by

Example 5.4: Consider a heated horizontal plate that is maintained at 100 °C. Air at 30 °C flows over with a velocity of 10 m/s. The plate area is 0.3 m2, and measurements reveal a drag force of 0.32 N on the plate by the flowing air. Determine the power (in W) that is required to maintain the plate at 100 °C.

Solution:

We use the modified Reynolds analogy to solve the problem. Air properties at 65 °C

ρ = 1.04 kg/m3 , ν = 19.51 × 10−6 m2 /s, k = 0.029 w/mK, Pr = 0.707, cp = 1.009 kJ/kgK

h =107.6 W/m2K

Q = hA(τ w − τ∞ )

Q = 107.6 × 0.3(70) = 2260 W

5.6 Internal flows 151

Please note that the use of the modified Reynolds number analogy is an absolutely clever way to solve the problem. In fact, the length scale in the problem is not given and so ReL and NuL cannot be defined, but then these hardly mattered. We used the original definition of Stanton number, St = h/ρu∞cp, to crack the problem. Additionally, from the friction data, we got the heat transfer. This is vintage engineering heat transfer.

5.6  Internal flows

Forced convection inside tubes and ducts has a very large number of applications, starting from the pipe carrying hot steam from the boiler to the turbine in a thermal power plant, to the tubes carrying the refrigerant in the condenser and evaporator of a refrigerator, and the tubes carrying the coolant in any automobile radiator or in any heat exchanger, for that matter.

The key challenge in the above mentioned applications is the calculation of the heat transfer coefficient h for the internal flow inside the tube in a particular application. Stated explicitly, if a constant heat flux or constant wall temperature on the outside of a tube (due to, say, vapor condensing on the outside), the key challenge is to get "h" for the fluid flowing inside the tube. In the more general case, along with this, the heat transfer coefficient on the outside, the conduction resistance across the tube material and any fouling resistance all have to be considered to obtain the overall heat transfer coefficient, which was briefly introduced in Chapter 1.

The key difference between external flows and internal flows is the concept of “developing” and “fully developed” flow, which can be best understood from a figure.

Consider a circular tube of diameter, d. A fluid enters the tube at x = 0 (see Fig. 5.8). Once the fluid enters the duct all over the surface, the presence of wall will be felt by the viscous fluid. Hence, a boundary layer starts developing as indicated in the figure. Once the boundary layer thickness δ becomes d /2, the whole of the flow inside the tube becomes a boundary layer flow, as opposed to the flat plate, where beyond y = δ in the free stream, the effects of fluid viscosity are not felt. The length X at which the whole of the flow becomes a full boundary layer flow is called entry length, and the flow beyond X, is known as “fully developed.”

FIGURE 5.8

Concept of developing flow inside a tube.

The geometry, coordinate axes, and the respective velocities are given in Fig. 5.9.

Comparing the scales of the two terms in the continuity equation, we have

Therefore, as x increases, v decreases. So, as we proceed deeper (i.e., into the fully developed region), v = 0.

From the continuity equation, in the fully developed flow region, ∂∂ux = 0.

Substituting for v = 0 in the r −momentum equation, with the understanding that v and all its derivatives are zero in the fully developed region, the r momentum equation reduces to ∂ p/ ∂r = 0. Hence ∂ p/ ∂x in Eq. (5.187) can be simplified as dp/dx . Eq. (5.187) now becomes

FIGURE 5.9

Problem geometry, coordinates, and velocities for laminar flow inside a pipe.